## Friday, April 28 — Holt 175, 4:00pm

### Carol Keig — Sonoma State Univ.

**Title:** The Poincaré Disk and the Hyperquilt: Easing into models of the hyperbolic plane

**Abstract:** In hyperbolic geometry, given line *l* and point *p* not on line *l*, there exist multiple lines through point *p* parallel to line *l*. Wait, what? This can be challenging to model or even to imagine. In this talk, we will briefly take stock of how we visualize Euclidean plane geometry, vanishing points, and spherical geometry, and then attempt to slide from there into visualizing models of hyperbolic geometry, starting with a partial model in the form of a homemade patchwork quilt, and moving to the Poincaré Disk model.

## Friday, April 21 — Holt 175, 4:00pm

### Kevin McGown — CSU, Chico

**Title:** Prime Numbers, Quadratic Forms, and L-Functions

**Abstract:** Does the sequence 2, 11, 20, 29, 38, 47, 56, 65, 74, … contain infinitely many primes? Which primes can be represented as *p*=*x*^{2}+*xy*+2*y*^{2}? We will discuss two classical topics in number theory -- primes in arithmetic progressions and representing primes by quadratic forms. We will see that the solutions to the problems mentioned above involve the study of zeta functions and L-functions, which are at the heart of the beautiful interplay between algebra and analysis found in modern number theory. No background knowledge will be required for this talk.

## Friday, April 14 — Holt 175, 4:00pm

### Guillermo Alesandroni — CSU, Chico

**Title:** The Magic of Steiner Triple Systems

**Abstract:** Steiner triple systems (STS’s) are a charming combinatorial topic consisting of points and sets of points that display certain symmetry. A distinctive mark of STS’s is the fact that its study feels like playing a game: a successful completion to the game generates an STS and, conversely, an STS provides a winning strategy to the game.

In this talk, we introduce STS’s as a combinatorial game and then, we give a tour through the main concepts and results on this area. Finally, we discuss some research projects that are within reach of students who may lack combinatorial expertise but have surplus of curiosity and enthusiasm.

Undergraduate students are especially encouraged to attend. In fact, this talk has been designed for them.

## Friday, Mar 24 — Holt 175, 3:00pm

### Gabriel Islambouli — UC Davis

**Title:** Multisections of smooth 4-manifolds

**Abstract:** Recently, Gay and Kirby introduced a new decomposition of a smooth 4-manifold called a trisection, which encodes the smooth structure as curves on a surface. We will discuss a generalization of this decomposition, which we call a multisection, and show that this generalization appears in many natural contexts, such as through generic paths of smooth maps of a surface to the real line. We also show how to carry out some important operations, including cork twisting and torus surgery, diagrammatically on these decompositions.

## Friday, Mar 3 — Holt 175, 3:00pm

### Jan Mazáč — Bielefeld Univ., Germany

**Title:** Patch frequencies in Penrose rhombic tiling

**Abstract:** In this talk, we introduce the Penrose tiling as well as an algebraic description of it using the so-called dualisation method for a certain four-dimensional lattice. We explain this method in two dimensions and, further, we extend it in four dimensions in a direct way. We show what the advantages of the method are and how one can use it for obtaining exact patch frequencies in Penrose tiling. This talk should avoid most of the technicalities, and is meant to be accessible to wide audience.

## Friday, Feb 24 — Holt 175, 4:00pm

### Michael Coons — CSU, Chico

**Title:** What we talk about when we talk about sequences

**Abstract:** Sequences are ubiquitous in mathematics, and since the advent of the digital computer, and with the more recent stress and importance put on data analysis, they have become ubiquitous in a much larger setting.

In this talk, I will discuss what we talk about when we talk about sequences… mostly from the point of view of number theory and dynamics, but to some extent in a more global picture. For example, how do we start to analyse a sequence? What do we need to do so? Are there other (easier) objects related to them? What kinds of conclusions can we make (e.g., quantitative vs. qualitative)?

This talk is meant to be accessible to a wide audience including undergraduates. I hope to see you there!

## Friday, Feb 17 — Holt 175, 4:00pm

### Daping Weng — UC Davis

**Title:** Cluster Algebras through Examples

**Abstract:** Since its introduction by Fomin and Zelevinsky in the early 2000's, cluster algebras have been observed in many different areas of mathematics, including combinatorics, algebraic geometry, low-dimensional topology, representation theory, and number theory. However, the definition of a cluster algebra is often found to be quite technical. In this talk, I will present some basic examples of cluster algebras to help make sense of the definition. I will also discuss some basic properties about cluster algebras as well as some open questions.

## Friday, Feb 3 — Holt 175, 4:00pm

### Maranda Smith — CSU, Chico

**Title:** Finding the Graded Characters of Demazure Modules of Current Algebras

**Abstract:** Since the 1980's, there has been high interest in studying finite-dimensional representations of quantum affine Lie Algebras. One way to study these modules tangibly is to take their graded limit, which converts these to representations of the corresponding current algebra. To better compare and identify these modules, it is helpful to know their graded characters, and so it is of interest to find closed forms of the graded characters of these modules. In 2015 Jeffery Wand introduced highest weight modules *M(ν,λ)* for type *A* which interpolate between level 1 and level 2 Demazure modules. In 2021 Biswal et. al. defined polynomials indexed by pairs of dominant integral weights, *G _{ν, λ}*(

*z,q*) where

*z*=(

*z*

_{1},…,

*z*

_{n+1}) ∈ ℂ

^{n+1}, and determined that

*G*(

_{ν, λ}*z,q*) has a closed form and is the graded character of

*M(ν,λ)*for compatible pairs (ν,λ). We construct analogs of these modules and polynomials for type

*D*. In this setting,

*M(ν,λ)*interpolates between level 1 Demazure modules and generalized Demazure modules as presented by Chari et. al. in 2019. We then create short exact sequences between these modules in order to identify the coefficients of the graded characters with the coefficients of

*G*(

_{ν, λ}*z,q*), proving that the graded character of

*M(ν,λ)*is

*G*(

_{ν, λ}*z,q*) for compatible pairs. In fact, this process provides a closed form for

*G*(

_{ν, λ}*z,q*).

## Friday, Jan 27 — Holt 175, 4:00pm

### Robert Lipshitz — University of Oregon

**Title:** Surfaces in 4-Space

**Abstract:** The goal of this talk is some recent results about the difference between smooth and non-smooth topology in four dimensions. We will start by motivating the distinction by thinking about curves in the plane and in 3-space. We will then discuss how to describe smooth surfaces in 4-space, and some longstanding open problems and recent results. Time permitting, we will end with some hints of how these results are proved.

This is almost entirely work of other people, though small parts are joint with Ozsváth-Thurston or Sarkar. At least the first 2/3 of the talk should be accessible to anyone familiar with continuous and differentiable functions of several variables.